How to rotate a bone around multiple times? It just snaps to closest rotation?
In the 3D View properties panel (activated with N), or in the Properties panel > Bone > Transform, switch the rotation mode from Quaternion to XYZ Euler, otherwise you will have the kind of problem you describe.

If you want your rotation to be linear (and not accelerate and decelerate, which will happen by default), go in the Graph Editor, select all and click T > Linear.

If you want the rotation to be infinite, select all and press shiftE > Linear Extrapolation.